Page 358 - A Course in Linear Algebra with Applications

342 Chapter Nine: Advanced Topics

Skew-symmetric bilinear forms
Having seen that there is a canonical form for symmetric
bilinear forms on real vector spaces, we are led to enquire if
something similar can be done for skew-symmetric bilinear
forms. By 9.3.2 this is equivalent to trying to describe all
skew-symmetric matrices up to congruence. The theorem that
follows provides a solution to this problem.

Theorem 9.3.6
Let f be a skew-symmetric bilinear form on an n-dimensional
vector space V over either R or C. Then there is an ordered
basis ofV with the form {ui, i , . . . ,u k, v fc, wi,.. .,w n _ 2 fc},
v
where 0 < 2k < n, such that

-
f(ui,v^ = 1 = /(vi,Ui), i = l,...,k
and f vanishes on all other pairs of basis elements.
Let us examine the consequence of this theorem before
setting out to prove it. If we use the basis provided by the
theorem, the bilinear form / is represented by the matrix

0 1 0 0 0 • • ° \
/
- 1 0 0 0 0 0

0 0 0 1 0 •• 0
0 0 • - 1 0 0 • 0

0 0 0 0 0 0

V 0 0 0 0 0 • 0 )

0 1
where the number of blocks of the type is k. This
- 1 0
allows us to draw an important conclusion about skew-
symmetric matrices.